Law of large numbers for Plancherel random Young diagrams

Do you know a reference book on the law of large numbers for random Plancherel Young diagrams ? I know the book of Kerov, but actually, it is only a compilation of his articles, and i need something more detailed.

Kerov-Vershik and Logan-Shepp independently proved in 1977 a law of large numbers for random Young diagrams ditributed according to the Plancherel measure. I am interested in the asymptotics and working on the article of Vershik-Kerov : "Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group" 1985.

In particular, I need for my master's thesis to control the $\| . \|_{\infty}$ norm of the difference between the limit shape and the normalized shape of a Young diagram with n boxes. There is an interesting result proved in the paper that I mentioned (Theorem 3), but I'm not convinced by the proof, because the authors assume that the shape has a support in $[-a, a]$ where $a$ is a constant. But in my opinion, the length of the support can be of order $O(\sqrt{n})$.

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